Exact Algebraic Filters and External Zeroing Operators

This paper establishes the foundational architecture of the External Zeroing Operator for continuous field equations. By relocating arithmetic selectivity from the internal oscillatory kernel to an external algebraic sensor Ψsensor(x)∈{0,1}\Psi_{\mathrm{sensor}}(x) \in \{0,1\}Ψsensor(x)∈{0,1}, the framework achieves exact binary topological phase partition: the integral retains full invariant energy when the sensor equals unity, or is annihilated identically when the sensor equals zero. Residual micro-energy arising from asymptotic Riemann–Lebesgue decay is eliminated at the algebraic level before quadrature. The core integral under the quartic damping kernel is shown to be a universal constant independent of the arithmetic selection mechanism.